Welcome to an introduction to infinite series.
The goals are to define a convergent and divergent
infinite series, as well as to determine
if a series appears to converge or diverge.
We will not be taking a look at specific
convergent or divergent tests.
Those will be covered in the next several videos.
The main purpose of this video
is to develop the idea of what it means
for a series to converge or diverge.
Since we just finished studying infinite sequences
it's important that we distinguish between
an infinite sequence and an infinite series.
Remember, an infinite sequence is just
an infinite list of terms
so if we take an infinite sequence an then sum the terms
we have what's called an infinite series, as we see here.
So to determine if an infinite series converges or diverges
we must consider the sequence of the partial sums
so what that means is, if we have an infinite series here
if we want to know if it converges
we want to take a look at the sum of the first term
which would be just one
the sum of the first two terms
which would be one plus four which equals five
the sum of the first three terms
which would equal 14, and so on
and the question becomes
does the sequence of the partial sums
one, five, 14, and 30, converge or diverge?
So the sequence is generated
by the partial sums of the infinite series.
It's probably pretty easy to see from this sequence.
It appears as though this is going to increase without bound
and therefore, this infinite series diverges.
We'll take a look at a more formal definition
in a few minutes.
Let's go and take a look at another infinite series.
Here we have the infinite series of
three divided by 10 to the power of n
so when n is one, we'd have
three divided by 10 to the first.
When n is two, we'd have
three divided by 10 to the second, and so on
and then we'd find the sum of these terms.
Let's take a look at the partial sums
in decimal notation.
The sum of the first term would just be
three divided by 10, or three tenths.
The sum of the first two terms would be
three tenths plus three hundredths
which would be point three three
and then the sum of the first three terms would be
three tenths plus three hundredths plus three thousandths
as we see here in decimal form
which is zero point three three three
and you can see the sum of the first four terms
would be zero point three three three three
so the question becomes
does the sequence created by the partial sums
as we see here, converge?
It looks like this three is going to repeat forever
and point three repeating should remind you
of the fraction one third
so it does appear that this infinite sum
will equal one third
and therefore, we say the series does converge.
Let's take a look at a few more examples graphically as well
by taking a look at this animation.
We're going to see the partial sums on the right
and then we'll see the graph
of the partial sums on the left
so as n increases from one to 10
we're seeing the graph on the left
and the actual parcel sums on the right.
It does seem pretty obvious from this graph
that it is increasing without bound
and therefore, we would say this series diverges.
Now this next series, generated by one over n
is called the harmonic series.
It's a very interesting series
and we'll take a closer look at this in a later video
but for right now, we're just taking a look
at the partial sums on the right
and the graph on the left
and it may seem unclear as to whether
this series converges or diverges.
We'll come back to this idea in another video.
I don't want to spoil all the fun here.
The next series we'll take a look at will be
one over n squared, and again
we'll be taking a look at specific tests
to determine whether a series converges or diverges
in the next several videos.
Again, looking at these partial sums
it does appear this series will converge
but again, we'll come back to that idea
in another video.
Let's go ahead and take a look at one more
where the terms in the series alternate sign
and again, we're looking at the graph on the left
and the partial sums on the right
so do you think this converges or diverges?
Well, we'll have to wait and see.
Let's go ahead and take a look at a more formal definition
of converging and diverging series.
For an infinite series, as we see here
the nth partial sum would be given
using this notation, where S sub n
will equal a sub one plus a sub two
all the way to a sub n.
If the sequence of the partial sums converge to L
we can say the limit as n approaches infinity
of the partial sums is equal to L
and if this is true, then the series converges.
We can also say the limit L
is the sum of the infinite series
and if S sub n diverges, then the series diverges.
I do want to mention one more idea for this video
and this is the divergent test for a series.
We will see this idea again in many of the other videos.
If the limit of a sub n, as n approaches infinity
does not equal zero, then the series diverges.
Remember, a sub n is the formula
that generates the terms in the series
so this is telling us that the terms don't approach zero
as n increases to infinity
then it's guaranteed that the series will diverge.
Hopefully, that makes sense
because there's no way an infinite series
can converge if we're adding numbers
that don't approach zero
so for example, if we have the infinite sum
of let's say, three n divided by n
we take the limit of a sub n.
This limit would equal three
and since this limit doesn't equal zero
we can say that this infinite series diverges.
Now, it's also true that if the limit of a sub n
doesn't exist, then the series diverges
so for example, if we had the infinite sum of just three n
well, we know the limit as n approaches infinity
of three n is equal to infinity
which doesn't exist, and therefore
this infinite series diverges
and then lastly, if the limit of a sub n equal zero
no conclusion can be made.
However, if the series does converge
then the limit of a sub n as n approaches infinity
must equal zero.
That's a little confusing, but what that's saying is
if we take a look at this harmonic series
we take the limit of a sub n
we do get zero, but that does not tell us
the series converges.
It tells us that it may or may not converge
but it is true that if it does converge
the limit of a sub n as n approaches infinity
does equal zero
so it's helpful in a way
that it tells us that it may converge
but it's not strong enough to tell us that it does converge
so that'll be it for this video.
We'll take a look at these rules again
and many other methods to determining whether
a sequence converges or diverges
in the next series of videos.
I hope you found this introduction helpful
and thank you for watching.